# Questions tagged [modular-curves]

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13
questions

**5**

votes

**1**answer

158 views

### Geometry of the section $X_0(N) \to X_0(pN)$ given by the canonical subgroup

Goren and Kassaei's paper "The Canonical Subgroup: a Subgroup-Free Approach" takes the position that the canonical subgroup of order $p$ for elliptic curves over $\mathbb{Z}_p$ with $\Gamma$-level ...

**2**

votes

**0**answers

142 views

### Standard application of Oort-Tate classification theorem

$\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\tors}{tors}$In Mazur's paper “Modular curves and the Eisenstein ideals”, on the bottom of page 159, it says that if $T$ is a open subscheme of $...

**2**

votes

**0**answers

102 views

### Weighted projective lines and elliptic curves

The modular curves of low level can sometimes be describes as weighted projective lines. For example, over $\mathbb{Z}[1/2]$ the compactified stack of elliptic curves with full level 2 structure is ...

**3**

votes

**1**answer

208 views

### Automorphisms of the modular curve defined over $\mathbb{Q}$

Let $p\geq 3$ be a prime number. The modular curve $X(p)$ can be considered as a connected smooth projective curve over complex numbers. There is a subgroup $\mathrm{PSL}_2(\mathbb{F}_p)$ inside its ...

**1**

vote

**1**answer

163 views

### When is $X_0(N)$ representable?

Fix a base ring $R$. Is the rigidification of the modular "curve" $X_0(N)$ in the sense of Abramovich-Olsson-Vistoli representable iff $N\geq 5$ and $N$ is $0, 2, 3, 6, 8, 11 (\mathrm{mod}\: 12)$? ...

**0**

votes

**0**answers

64 views

### Isomorphism between 2nd symmetric product and Jacobian

Let $X=X_0(N)$ be hyperelliptic with $g(X)\geq 2$ with $\infty$ as a cusp and $\iota$ as the hyperelliptic involution. Then the map
$$X^{(2)} \longrightarrow Jac(X)$$ $$D \longrightarrow [D-\infty -\...

**1**

vote

**0**answers

90 views

### What is the modular curve for level 1, 2?

An elliptic curve over a scheme $S$ is the data of a proper smooth morphism
of schemes $\pi: E\rightarrow S$ whose geometric fibres are connected curves of genus $1$ and a section $e: S \rightarrow E$....

**3**

votes

**1**answer

173 views

### When is the conductor of an elliptic modular curve equal to its level?

Suppose the usual modular curve $E=X_0(N)$ over $\mathbb{Q}$ has genus 1 (e.g. $N=15$). Define the conductor of $E/\mathbb{Q}$ as the ideal/integer:
$$M=\prod_{p}p^{f(E/\mathbb{Q}_p)},$$
where
$$f(...

**2**

votes

**0**answers

102 views

### Local-global compatibility and modular curves

I have been told by some people that local-global Langlands compatibility for $GL_2$ (the vanilla version, not the one being developed in this decade by Emerton and others) implies Shimura conjecture ...

**5**

votes

**0**answers

257 views

### Integral models of perfectoid modular curves

Scholze constructed perfectoid modular curve and its canonical and anticanonical part in his paper On torsion in the cohomology of locally symmetric varieties (Annals of Mathematics 182 (2015) pp 945–...

**2**

votes

**0**answers

119 views

### Igusa curve at infinite level

In the paper "Le halo spectral" (http://perso.ens-lyon.fr/vincent.pilloni/halofinal.pdf) and in the following paper "The adic cuspidal, Hilbert eigenvarieties" (http://perso.ens-lyon.fr/vincent....

**7**

votes

**3**answers

415 views

### Endomorphism ring of $J_0(p)$ and Hecke operators

Let $J_0(p)$ be Jacobian of the modular curve $X_0(p)$ over $\mathbb Q$ where $p$ is a prime, consider the subring $\mathbb T$ inside $\newcommand{\End}{\operatorname{End}}\End_{\mathbb Q}(J_0(p))$ ...

**10**

votes

**0**answers

217 views

### Generalized Jacobians and modular units

Let $X$ be a proper algebraic smooth curve over a characteristic zero field $k$ and let $J$ be the Jacobian variety of $X$. Let $K$ be the function field of $X$. Assume that we are given $n$ distinct ...